Some remarks on injective envelopes on ring extensions

TL;DR

This paper introduces and investigates a new class of injective modules related to ring extensions, establishing their properties and existence under certain conditions, such as when the subring is pure-semisimple.

Contribution

It defines $(R, S)_\star$-injective modules and proves the existence of their envelopes in the context of ring extensions, especially for pure-semisimple subrings.

Findings

Defined $(R, S)_\star$-injective modules and studied their properties.

Proved the existence of $(R, S)_\star$-injective envelopes.

Established that every $R$-module has an $(R, S)$-injective envelope when $S$ is pure-semisimple.

Abstract

Let $f:S\rightarrow R$ be a ring extension. We introduce and study the properties of $(R, S)_\star$-injective modules and the existences of $(R, S)_\star$-injective envelopes. Besides, we show that every $R$-module has an $(R, S)$-injective envelope when $S$ is a pure-semisimple ring.